7 THE STURM-L1OUVILLE PROBLEM 355

We can in addition suppose that at the points a, b, the value of wm is 0 or 1; it
is then clear that \u(x) - wm(x)\ < \\u\\ 4- 1 in each of the intervals

0} a -|— an(i ft— 6
L wj L rn J

and therefore \\u — wm\\2 is arbitrarily small by the mean value theorem; as
wm satisfies all conditions in (d), this proves our assertion. Once (c) is thus
proved, it is clear that the total sequence (cpn) must be infinite, and (applying
(11.6.2)), (a) is also completely proved. Finally, (e) and (f) follow at once from
(11.5.11).

Remark. It is possible to obtain much more precise information on the <pn and
\n, and to prove in particular that kjn2 tends to a finite limit (see Problems
3 and 4).

PROBLEMS

I. Let I = [a, b] be a compact interval in R, and let H0 be the real vector space of all
real-valued continuously differentiable functions in I; H0 is made into a real pre-
hilbert space by the scalar product

(a) Show that Ho is separable (approximate the derivative of a function x e H0 by
polynomials (7.4.1)); H0 is therefore a dense subspace of a separable Hilbert space H
(6.6.2).

(b) If (xn) is a Cauchy sequence in the prehilbert space HO , show that the sequence
(xn) is uniformly convergent to a continuous function v in I, and that if (yn) is a second
Cauchy sequence in H0 having the same limit in H, then (yn) converges uniformly in I
to the same function v; the elements of H can thus be identified to some continuous
functions in I, which however need not be differentiable at every point of I. (Observe

U& \l/2

x'2 dt\ in I.) Show that,

for any function z e H0 which is twice continuously differentiable in I and such that
z'(a) = z'(b) = 0, (v | z)« — f * vz" dt -f f * vz dt.

(c) Let a, ft be two real numbers, q a continuous function in I. Show that in H0, the
function x -* ®(x) « P (x'2 + gx2) dt - oc(x(a))2 - fi(x(b))2 is continuous. Let A be the

f b
subset of H consisting of the functions x such that x2 dt— 1 (observe that this is not

a bounded set in the Hilbert space H). Show that in A n H0, the g.l.b. of ®(x) is finite.
(One need only consider the case a>6,')8>0. Assume there is a sequence (xn) inhe boundary conditions (11.7.2).